| In this talk, we will elaborate the importance of the study on the behavior of high frequency components in a solution to a partial differential equations. We will explain why the understanding of high frequency behavior is crucial for designing efficient discretization methods for PDEs and for designing efficient solver for the discretized systems. We will illustrate that high frequency components tend to behave locally for a large class of partial differential equations. For elliptic partial differential equations, we will prove the point-locality for high frequency components. As examples of applications, we will talk about adaptive mesh refinement, domain decomposition, multigrid methods and other special and efficient discretization and solution techniques. |